Volume 4, issue 1 (2004)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 23
Issue 9, 3909–4400
Issue 8, 3417–3908
Issue 7, 2925–3415
Issue 6, 2415–2924
Issue 5, 1935–2414
Issue 4, 1463–1934
Issue 3, 963–1462
Issue 2, 509–962
Issue 1, 1–508

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
The boundary-Wecken classification of surfaces

Robert F Brown and Michael R Kelly

Algebraic & Geometric Topology 4 (2004) 49–71

arXiv: math.AT/0402334


Let X be a compact 2-manifold with nonempty boundary X and let f : (X,X) (X,X) be a boundary-preserving map. Denote by MF[f] the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to f. The relative Nielsen number N(f) is the sum of the number of essential fixed point classes of the restriction f̄: X X and the number of essential fixed point classes of f that do not contain essential fixed point classes of f̄. We prove that if X is the Möbius band with one (open) disc removed, then MF[f] N(f) 1 for all maps f : (X,X) (X,X). This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If X is the disc, annulus or Möbius band, then X is boundary-Wecken, that is, MF[f] = N(f) for all boundary-preserving maps. If X is the disc with two discs removed or the Möbius band with one disc removed, then X is not boundary-Wecken, but MF[f] N(f) 1. All other surfaces are totally non-boundary-Wecken, that is, given an integer k 1, there is a map fk: (X,X) (X,X) such that MF[fk] N(fk) k.

boundary-Wecken, relative Nielsen number, punctured Möbius band, boundary-preserving map
Mathematical Subject Classification 2000
Primary: 55M20
Secondary: 54H25, 57N05
Forward citations
Received: 21 November 2002
Revised: 15 October 2003
Accepted: 26 November 2003
Published: 7 February 2004
Robert F Brown
Department of Mathematics
University of California
Los Angeles CA 90095-1555
Michael R Kelly
Department of Mathematics and Computer Science
Loyola University
6363 St Charles Avenue
New Orleans LA 70118